Kodachrome did not become obsolete. It was discontinued. Big difference. There is still no other film emulsion that has the archival characteristics of Kodachrome. In my opinion, its discontinuation has left a big hole in film photography, and in fact in photography in general, which may never be filled. I hope I will be proven wrong some day but that's the way I see things now.
So because of Kodachrome's unique characteristics, attempting to "replicate" it may not even be possible. But you can certainly get close. Current E6-based slide emulsions have a different look than Kodachrome, but if you use a well-balanced one, at least your results can be neutral. The two best that are available right now -- IMO -- are Fuji Provia 100 and Fuji Velvia 50. Velvia's grain structure certainly rivals Kodachrome's but Velvia can be rather wild in the way it saturates colors. Provia is more neutral in that respect, although its grain is more obvious. Even so, Provia's grain is still very fine, so it might be the best emulsion to use. Kodak's Elite Chrome 100 was really good stuff -- at least as good as Provia 100, I feel, but it's been discontinued for a while now. You can find quantities of it for sale on eBay still, so it's still out there. Most is probably expired by now, but if it's been kept frozen, or at least refrigerated, it should still do a good job.
So there's two possibilities for you -- three if you want to give Velvia a try. Time to start shooting and see how close this modern E6 stuff compares to the old Kodachrome photos you have.