I don't mean to be a narrow-minded person; power chords are quite good. They're possibly one of the best chords invented for guitar, and they certainly sound nice. It's just that overuse of them in songs nowadays (and even in the older days) has led to their use becoming really unoriginal.
Now, don't get me wrong, there's nothing really wrong about power chords. Go ahead, use them if you want, but I just want to emphasise the fact that overuse of them will get boring and quite monotonous. Besides, there are plenty of other types of chords you could put in place of a power chord and it would still sound relatively the same, though in most cases it actually sounds better. Here are a few things you could just pop in there instead of a power chord and it would still sound great.
Since a power chord consists only of the root note, the fifth, and occasionally octave, the chord simply cannot be major or minor. If you want your song (or just a part in your song) to specifically sound major or minor in key (e.g. major if it is seemingly happy or uplifting, minor if it should be depressing in some way), then it would be much better to play a full major or minor chord. For instance, if you want something to sound specifically C major, then just play a C major chord. A C power chord won't quite sound either major or minor, it will be teetering on the fence, but that extra stress of major will be enough to push it over the edge. Though in most cases the full chord is harder to play (especially to form the chord shape in time), but this is quite worth it.
Perfect Fourth Double-Stops
What exactly is a perfect fourth double-stop? It's when you just play the root note and then the perfect fourth, at the same time. For instance, sticking to C, a C perfect fourth double-stop would call for a C and then an F. So, in other words, a 5th string 3rd fret and 4th string 3rd fret is a perfect fourth double stop.
Anyway, these really spice up your song. If your style is metal, then double-stops such as these go great in your rhythm guitar parts. They provide a much "heavier" sound than a power chord. Some great examples of good use of perfect fourth double-stops in metal would be the main riff for Metallica's "Creeping Death" and the opening riff for Iron Maiden's "2 Minutes To Midnight". These double-stops also sound good in solos, particularly anything country and blues. If you're doing, for instance, a hammeron in between two notes, just whip a perfect fourth up above your first note and hold it when you hammeron to the next note. Sounds great in most cases.
Move the Bass Line While Keeping The Same Riff
There's nothing worse than an extremely repetitious riff played over and over and over for an unnecessary amount of time. That being said, don't include a really great riff just once or twice, because that leaves the listener wanting more and then feeling disappointed. If you do happen to have a really great riff, it's nice to repeat it a while; in fact, it may be crucial to repeat it because most listeners will like it. So how do you repeat a good riff and not make it get old and sound monotonous? Move the bass line around. You will still keep the same riff but it will sound a bit different if, say, the bass goes down two steps, comes back up one, and then goes back to the original bass line. Iron Maiden are prime examples for this, and this technique is used in a mass number of their songs. To name a few, "The Trooper", "Aces High", and "Die With Your Boots On". So, if you're ever stuck with a great riff that repeats twenty times before your verse or something (or, in most cases, if it's for the intro), just move the bassline around each time you repeat the riff, so the riff will be the same, but it won't always sound exactly the same because of the bass.
Any riff can pretty much stand on its own, but it never hurts to harmonize. In many cases harmonizing can take a riff to new heights, and really make it shine. When harmonizing, experiment harmonizing with fifths, fourths, major and minor thirds... just about anything. They sound so much better than just one set of notes. Once again, Iron Maiden were probably the pioneers of this concept... either them or Thin Lizzy. Speaking of Thin Lizzy, think about the harmony interludes in between each chorus in "The Boys Are Back In Town". Think how bland they would have sounded if just the lower notes or just the higher notes had been played. Sure, it still would have sounded decent, but harmonizing it really makes it sound a hundred times more interesting. The great thing about harmonizing is it can be used in both rhythm parts and lead parts. Doing rhythm in lead parts can be strenuous, because you'll probably have to have two guitarists play two different sets of notes and stay in perfect in-synch tempo. Overdubs, however, can be done during the recording process and are probably much easier than trying to get both guitarists to play every note at just the right time. But when you pull it off, harmonizing sounds beautiful.
Inverting a chord is exactly how it sounds: invert the order of the notes. Staying in C, the notes of a C chord are CEG. If you play any other combination of those notes (EGC, CGE, GEC, GCE, EGC), you're still playing a C chord, only the different combination of notes can sometimes sound quite a bit different than the original combination. Experiment with them sometimes, you may just find a great combination.
Whether you're a beginner guitarist or not, learning first-position chords is important, no doubt. But later down the road you're going to find that different chord shapes form interesting variations to just the regular first-position chords. Try each root note higher up on the neck, trying to map out where each next note will fall. Since you know a C chord is CEG, you know you need to find a C root note. Here is a lits of every single C root note on the neck:
1st string: 8th fret, 20th fret
2nd string: 1st fret, 13th fret
3rd string: 5th fret, 17th fret
4th string: 10th fret, 22nd fret
5th string: 3rd fret, 15th fret
6th string: 8th fret, 20th fret
Knowing that those are all the root positions for your C chord, you can use any one of those as your root note and try to find the E and the G that goes along with it. And, of course, E and G could also be your root note, just so long as you stick with CEG, so you can start on any one of the CEG's on your fretboard and form the other two notes and you've got your C chord. Countless combinations. A good example of different voicings would be the intro to Megadeth's "Hangar 18". It starts off with a chord progression of D minor, D minor #5, D minor 6, and A minor add 11. Only after those four chords are played, they are played again in a different chord variation, giving them a higher-pitched sound. Whereas just playing those four chords twice for one phrase, they were translated into different voicings to make them sound different, yet they are the same chords. Pretty amazing, huh?
Well, those are the main things you can do instead of a boring old power chord. Of course, there are many others, such as barred chords, chords with different notes in the base (or bass? Or both?), augmented and diminished chords, flatted/sharped chords, usage of different intervals in double-stopping, and maybe the verse in major and the chorus in minor or something like that. Hey, it's your song, do with it whatever you want. Just letting you know that some of the greatest and most influential bands of just about any given era would have thrown up at the thought of constant power chords.