I've used Morgans Kits and haven't had that 'kit twang', but because they are more heavy in flavour than the equivalent Coopers kits, I have made a couple of horrendous mistakes by trying to beef them up with specialty grains and extra hops and ended up with excessively dark, muddy and eye-poppingly bitter beers. Some of their tins like Royal Oak Amber and Australian Draught are best left alone in the hopping and dark malt department - they can stand on their own two feet.
If using a kit now, I tend to go for the lightest possible - for example a Morgans Canadian or a good old Coopers Lager original series. This gives me far more leeway to add subtle amounts of hops, steeping grains etc. And I always chuck the kit yeast or use it for ginger beer. I find that Nottingham does a great job and settles out well, and recultures well. Also the temperature is important, I try to keep below 20 degrees which is hard in SEQ at the moment.
//beginning of rant <_<
IMHO (dons flameproof suit) another source of 'twang' is excessive use of light dried malt extract. Lets face it, LDME wasn't invented for the benefit of us home brewers. Whilst nowadays of course, a fair amount is used by us, the majority goes into things like Milo bars, Sustagen gold, Maltesers, Sarah Lee cakes, Mars Bars etc etc.
It's a lot cheaper than buying kilo or kilo and a half tins of liquid malt extract from Coopers or Morgans, but I'm becoming increasingly iffy about LDME. The best received and cleanest K&K I ever made was brew number one in June that was a Morgans Queensland Pilsener and a can of Morgans light malt liquid and a Saaz teabag. And it was cool at the time, fermented out at around 16* And it cost nearly $30 which I consider rude for a K&K
Recently I made two brews a few weeks apart, both of them with a coopers lager tin, same hops, same yeast (Notto). In the first one I used heaps of LDME, in the second one I did a partial mash with just an English (Bairds) pale malt and added dreaded cane sugar instead of the LDME and it turned out a really nice beer with no 'cidery' twang, which I had feared.
In comparison the LDME brew is muddy, dark - a Coopers Lager should not turn out the colour of an English Burton bitter. (I think the 'L' in LDME varies from batch to batch)
The use of the sugar suited the particular style of the beer but I wouldn't even contemplate it with dark mild or an English bitter etc as it would definitely thin out the beer and stand out like the dog's proverbials.
//
end of anti LDME rant. 
However at the end of the day as Reviled said the way to completely dispense with home brew twang and set yourself freeeee is to go AG, as I am currently doing.
