Houses And Apartments For Rent In Brownsville Tn . 1282 fairground rd, brownsville, tn 38012. See all 4 houses in brownsville, tn currently available for rent. 3367 Brownsville Rd, Clarksville, TN 37043 Trulia from www.trulia.com Houses for rent in brownsville, tn. An apartment home for rent in this town costs renters from $380 to $1,885. In terms of housing, residents are fairly evenly split between renting (51%) and owning (49%) their home.