First off are the groups that you are looking for Security or Distribution groups? If they are distribution groups they won't be valid for use by SharePoint.
Next is what domain they reside in and where the users reside in. If SharePoint and the users originally resided in domain A and both have moved to domain B then this may not be necessary. However if SharePoint is still in Domain A then this will be necessary. A good way to test this is to see if a new user is able to be queried from the new domain. If you can't get items that don't already exist within SharePoint via people picker from a certain domain the cause is likely either that you are not querying that domain or that there is a permissions or communication problem when performing the query to that domain.
The permissions can be rooted out relatively quickly. SharePoint uses the identity of the application pool that the IIS site is running as to query the domain UNLESS otherwise specified. Make sure that this account has permissions to AD to do this and if not then either grant those permissions, change the user, or specify a different account to be used to perform the queries.
Run the below command to see if there are any configurations. This will tell you if the web app is specified to do anything other than the default behavior. Meaning that if this is blank it should be querying the domain that the server belongs to only. If there is a value here it will follow what the settings are.
stsadm -o getproperty -pn peoplepicker-searchadforests -url "http://webapp"
If there the new domain isn't listed here then you will want to specify the domains that will be queried for this. The below command is an example of how you would structure the command to set up a web application to query multiple sources.
stsadm -o setproperty -pn peoplepicker-searchadforests -pv "domain:contoso.com,username,password;domain:fabrikam.com,username,password" -url "http://webapp"
Note that the username and password are optional. Also by default SharePoint only queries itself and the domain that it is a part of. It could be possible that you are not locating anything from the new domain because all existing users will already exist in the UserInfo table which is the first location that SharePoint looks to.