I agree with bart59. Properties along Sheik Zayed Road are nice, this is a high-end area. You could also check Burj Dubai, Dubai Marina, Jumeirah Beach Residence, Jumeirah Lake Towers & Palm Jumeirah. Those areas are really nice.
Average Rent of Dubai Apartments within those areas will range from US$1,500 above. But you can still find cheaper rents around those areas, it will depend on your preference.