Well you could take a look here:

http://www.megalithomania.com/

Unfortunately you are staying in the wrong place to megalithic sites: the Midlands are pretty barren. Depending upon what you call a day trip I would recommend

Heading north:

Cohaw (Cavan)
Beaghmore (Tyrone)
Creggandevesky (Tyrone)
Drumskinny (Tyrone)

Heading west:

Carrowmore (Sligo)
Annaghmore (Leitrim)
Loughscur (Leitrim)

If you want to head east/northeast:

Proleek (Louth)
Clontygora, Ballymacdermott, Ballykeel (Armagh)
Legananny (Down)

From Cavan you have a long way to go if you want to head south.

I would recommend looking at these sites, looking at what's close by and picking a direction. Then buy the OS maps for the area and take a look at what else is around. Most sites in Ireland are on private property, but all the ones listed above are definitely public access.

If you have any more specific questions feel free to ask.