If your RDBMS supports them and if you use them correctly (and consistently), unique keys on the composite PK should be sufficient to avoid duplicates. In SQL Server at least, you can also create FKs against a unique key instead of the PK, which can be useful.
The advantage of a single "id" column (or surrogate key) is that it can improve performance by making for a narrower key. Since this key may be carried to indexes on that table (as a pointer back to the physical row from the index row) and other tables as a FK column that can decrease space and improve performance. A lot of it depends on the specific architecture of your RDBMS though. I'm not familiar enough with Access to comment on that unfortunately.
As Quassnoi points out, some ORMs (and other third party applications, ETL solutions, etc.) don't have the capability to handle composite keys. Other than some ORMs though, most recent third party apps worth anything will support composite keys though. ORMs have been a little slower in adopting that in general though.
My personal preference for composite keys is that although a unique index can solve the problem of duplicates, I've yet to see a development shop that actually fully used them. Most developers get lazy about it. They throw on an auto-incrementing ID and move on. Then, six months down the road they pay me a lot of money to fix their duplicate data issues.
Another issue, is that auto-incrementing IDs aren't generally portable. Sure, you can move them around between systems, but since they have no actual basis in the real world it's impossible to determine one given everything else about an entity. This becomes a big deal in ETL.
PKs are a pretty important thing in the data modeling world and they generally deserve more thought then, "add an auto-incrementing ID" if you want your data to be consistent and clean.
Surrogate keys are also useful, but I prefer to use them when I have a known performance issue that I'm trying to deal with. Otherwise it's the classic problem of wasting time trying to solve a problem that you might not even have.
One last note... on cross-reference tables (or joining tables as some call them) it's a little silly (in my opinion) to add a surrogate key unless required by an ORM.