You sure you're actually not getting the achievements? Your notifications could just be set to off, so press Xbox Home button, go to Achievements and see if you're actually getting the achievements. If you aren't getting the achievements, then check to see if you're actually playing on Normal, or if you already got the achievements.
If all these things aren't the case, I have no idea.
There's a common issue with the 'If They Came To Hear Me Beg' achievement not popping up the first few times. Happened to me too. Haven't heard of any other problems, though. Like Luigi said, you might have turned off your notifications. If you really didn't get them, I'd give Customer Support a call. They might know what's up.
If They Came to Hear Me Beg was a bitch. It took me over a year to get the achievement, after properly executing the challenge multiple times at both the last mission, and from the top of the spire. Other than that, Reach's achievements have worked fine for me.