I usually do my prints at the Lab where a Fuji Frontier is used.
For the most part, my prints are done using Matte. In my opinion the quality of prints is very nice.
I would like to know if it is possible to achieve the same quality as the Frontier using any of the newer inkjets i.e. Canon IPF5000, HP9180, Epson 4800.
If so, what type of paper should I be using, Matte or Luster?
I think the Fuji Crystal Matte paper is more like semigloss or luster surface in inkjet terms; matte inkjet papers are completely different (although nice in their own way). To get the "look" of the prints you're used to a luster paper that isn't too "toothy" would probably be a good match, such as Ilford Smooth Pearl.
As to the quality issue, today inkjets are better than the Frontier lab prints by pretty much any measure you can think of: resolution, color gamut, dmax, even archival rating if you use pigment inks.