That's the Dundee neighborhood. Our first apartment was at 50th and California, about five(?) blocks north of Dodge. (The main E-W drag through Omaha)
Edit: And GWad beat me to it.
Again, it depends on where you would be working. If you're working downtown, I really wouldn't look any farther west than 90th St. You would be better off looking at Bellevue/Papillion/LaVista, since they're pretty much directly south of downtown.
Omaha is, I believe, a pretty nice city. I wouldn't mind moving back, but it would put us farther away from both of our families. If you're a baseball fan, and do move out there before mid-June, make sure you get to a couple of the College World Series games at the new ballpark downtown. The atmosphere is a touch more sanitized since they moved out/tore down Rosenblatt, but it's still a great experience.
Yep I lived in a brick duplex 1 block south of Dodge on 50th. I seem to remember a school or or something just to the east of my place? I can't remember the cross street that went e/w just south of Dodge. Anyway it was a great neighborhood to live in.