Check in settings under notifications. There is some control on the Droid 1 for what ways the notifications are done. I am sure it will have something there for the X. My X hasn't shipped yet, so I can't say for sure.
Another option is to enable the data timeout under Settings > Battery Manager > Battery Profile Options. You can tell the phone to turn off data after X minutes of no activity during certain hours. I have mine set from 10pm - 5am and so far the phone is silent during those hours unless I turn it on for something. I can still get phone calls and the alarms work just fine too with data off. Hope this helps.