We stayed at the Alexandra Guest House when visiting Limerick. The guest house was about 15 minutes walk from the city centre, on O'Connell Avenue - at the end of O'Connell Street - one of the main streets in Limerick.
Our room in the guest house was clean and well decorated. We had one of the rooms with an en suite (shower and toilet), and I think there are other rooms with shared bathroom facilities at a lower price. Also included in the price was a full breakfast.
We payed about 35 Euros per person per night for a room with en suite, which seems good value.
If you are looking for inexpensive, clean accomodation close to the centre of Limerick then this is a good place to try.
The only problem we had was finding the place, not because it isn't centrally located but because Limerick has a "mad" one way system so it may be an idea to get some directions when booking a room so you don't spend as much time as we did driving round the city!!!