Are your push notifications not coming through on Android?

Here are a few steps that might fix the issue:

1. Double check that you have the Android Market or Play Store on your phone AND that your version of android is 2.2+. If your phone doesn't have the android market (or you're using android v2.1), then just wait for the next groupme update (v3.3.2 which will have the push-server stuff) and reliability should be greatly improved. 
2. Make sure that you have a valid google account synced to your phone. Do this by going to Android's settings -> Accounts (or Accounts & Sync depending on your version of android). If you don't see a google account, then add one (you don't actually need to sync gmail or contacts or anything, but you must have a google account). If your google account has a little red exclamation point next to it, then you most likely need to update your password on your device (which you should be able to do from this screen).
3. If you are in the USA, go into GroupMe's global settings and check, then un-check the SMS mode option
4. If you are not in the USA, then goto groupme settings and click Reset App.