Hmm, here are a few. There's no accounting for taste, this is just what I found after a quick skim through my MP3 folder. ;)

"Accidentally in Love" by Counting Crows

"Heaven" by Bryan Adams (covered by DJ Sammy/Yanou if you want female vocals. They have an upbeat dance version and a slow "Candlenight" mix)

"Ain't No Mountain High Enough" by...well, whichever version you like best. :)

"I Say a Little Prayer" by Dionne Warwick or Aretha Franklin

"With You" by Jessica Simpson (first verse has references to the singer being female, which may mess up the gender-neutralness)