I like them in mixed borders with other perennials.. and the variety of perennials they rub shoulders with is really too big to get into here! (We can grow an amazing range of hardy perennials in Calgary, where I see you are situated too.) So don't limit yourself. Compatible conditions, heights and avoiding over-crowding are about the only rules.
Also, lilies don't really die back until very late in fall, so there is really no need to be concerned about "hiding dying foliage". (Perhaps you are thinking more of bulbs such as tulips, daffodils that go dormant after blooming?) If you select your lily cultivars and species for the widest bloom range, you can have lilies flowering from June into September. And in fact, very late in fall, lilies provide some very nice fall color, as the stems and leaves on different varieties turn various shades of yellow, gold, burgundy and purple. As botrytis is not a concern in our very dry climate here, I have never seen any need to cut down lily stems, thus they add interest through the winter too.

I use lilies everywhere.

Mixed asiatics with Cinnamon Fern

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Endless Summer Hydrangea and Pink Lilies

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Trumpet Lilies near Crab Apple Tree and hostas

Daylily and White Henryi, purely by accident

Daylilies, Catmint & LA Hybrid Lilies

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My spring tulip garden gives way to a garden full of lilies & glads with a spattering of peonies in-between the clumps. The image below has some lilies that I have no clue as to their names, however the spent tulips foliage can't be seen - too bad the place is being overrun by red lily beetles.
First come the tulips.....
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Then come the lilies....
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Nothing much left to see from the original tulip foliage...
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I like the combination of warm-coloured lilies (such as 'Grand Cru', or 'African Queen') with large-leafed foliage plants such as Canna, Ricinus, bananas or Fatsia.

As shown in Leslie197's picture, lilies can also look great with large ferns.

Bright-coloured lilies surrounded by Queen Anne's Lace.

White or pastel lilies growing up through a cloud of Rozanne geraniums.

Just about any kind fronted by a fat clump of heuchera. Palace Purple in front of pale Oriental lilies is very nice.

Last year, 'Black Beauty' was beautiful alongside Echinacea 'Magnus', with some arching branches of 'White Dawn' rose above.

- Steve