DaveHax made one of the best videos (top in this step) about the Ukrainian Lacing so far. Check yourself.

Look, how the shoelace ends are fixed in the bottom pair of eyelets in the video. This is just one of SEVERAL ways of doing that. Another possible way - when fixing the shoelace ends in the bottom pair of eyelets you might try to pass it a second time through the OPPOSITE eyelet, not the same. Dave's way is great, but in some cases (e.g. a soft edge of the shoe, too long shoelaces, visual preferences) the above can also serve well. And, of course, you might tie the shoelace ends together with a neat knot of your choice or use a clip for the same purpose.

You might also prefer the T-Laces (shown on the picture above), though "no special shoelaces needed" approach seems to be nicer in some regard. BTW, I use knots as shown in the instruction above and have never felt them, though, sure, it depends on the particular shoes and shoelaces (very thick shoelaces might be a challenge in this regard indeed).

Moreover, you can omit the "fixation tricks" altogether and just leave the aglets behind the bottom eyelets if the eyelets are relatively small like it's shown in a nice video by Itty Bitty Pals (above). When the shoes are put on it's most unlikely that the aglets will slip out the eyelets in this case.

UPDATE: Ian Fieggen (author of the famous Ian Knot) has added a great tutorial about the Ukrainian Lacing to the best ever online shoe lacing encyclopedia, Ian's Shoelace Site: www.fieggen.com/shoelace/ukrainianlacing.htm