Mondo grass would be a good bet if you want something low growing that resembles grass.

How much shade/sun do you get here?

If you want something that flowers, try Vinca (Periwinkle) for part shade.

Dwarf perennial chamomile?

Creeping or wooly thyme?

Irish or Scotch moss are lawn looking & used in Japanese gardens. I like how easily it is spreading at the edges of the gravel driveway. Sure beats weeds!

There are a few groundcovers that I've been using in my part sun/shade areas, though none are grass like.

spotted dead nettle (Lamiums)
golden creeping Jenny (Lysimachia)
creeping Veronica ped... 'Georgia Blue'

Lamium mac 'Aurea' on the right (pointed bright green leaves w/ pink flowers)
golden creeping Jenny on the left (round leaves)
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My top pick that puts up with foot traffic is Ajuga reptans & I like the way it's filled in between stepping stones near my hose reel. It isn't fussy about soil & seems to spread more if moist, but does okay with less water if you mulch well. You might find out how it does locally for you on a regional forum.

In this 7a area I find Mazus repens does well, and it stays green, all Winter. Ajuga is tough, evergreen, here and only needs to have spent flower heads sheared off once a year.
But I do pull tree seedlings, etc. from the rug-like ground-covers from time to time.
Mondo grass would be much more costly, as I don't think it would spread like the two plants above, but it is beautiful and seems weed omitting.