MIUI's built in SMS + MMS backup works quite well. However, I want to try out new custom ROMs. Is it possible to add integration of that MIUI service onto another ICS rom?
I don't think you can use the MIUI specific functionality, but using Titanium backup you can backup your SMS and MMS:
Now Titanium Backup will automatically backup your messages at the specified time and day or days.
Note: You will need the Titanium backup pro key to access the batch operations.
You can also backup your call call history by adding '[CONTACTS/CALLS] Contacts Storage 2.2' or 'Call Log' to a label
Taken in part from this guide on XDA
You can backup SMS, MMS and call logs to your GMail account with this free app. Being online, you wouldn't lose your backups even if wipe your sd card or lose your phone.
This is better than using Titanium Backup, as it can sometimes cause issues.
Answer: No you can't.
Reason: You can't install the MIUI Backup App on another rom, because it's a signed system APK.
Solution: Use a backup solution which is available for all roms. If you just want to backup and restore SMS, then use the free https://play.google.com/store/apps/details?id=com.riteshsahu.SMSBackupRestore
Source: I'm a MIUI Beta tester and former moderator on the original forum en.miui.com