If you looking for a natural look, you could put round stones along the edge. That will help to keep the pipe on hold to the round and make it less visible.

Do you have metal or plastic edging along the edge line? They helps in keeping the 2-inch mulch run off and make the edge line more defined.

To me this is an easy fix. Your landscaper was going for an "English" or trenched edge to the beds. When done properly and maintained well, this is a gorgeous, clean, classic look. To clean up the bed edges you can use a manual edger or you can use a shovel (straight down at the edge to make a new, clean cut in the sod).
In order to cover the drainage pipe, just bring that bed edge out a few more inches and cover it up more with mulch. The dark mulch was a good choice especially as the black drainage pipe will disappear into it. I wouldn't be averse to actually extending the pipe even further from the house if possible/practical, either, but that might just be my own water issues talking! If you look carefully at the level of the lawn just in front of the end of the pipe, it's actually higher than the outlet of the pipe, so water could potentially flow back INTO your bed and toward your house, which you do not want. Try to make sure you give the water a lower spot to run towards.

I was googling edging materials the other day and came across a product called "The Edge". When it's installed you can't see the edging and it gives a really clean, nice look. I haven't tried it and have no idea if the product is any good but the pictures are enticing me to try it. Anyone else out there heard of this?

If the drain is from a gutter downspout, it would be best to relocate it or extend it outside the mulch bed because it will wash the mulch away in a rain. If it is a french drain, it can be covered by mulch.
For a clean looking edge, especially if you have bermuda grass, several products (steel, composite and black plastic)are made that work well if installed properly. They all are dug in about 5-6" and should be less that an inch showing above the ground level. The grass will hide the edging and dark mulch blends with the black plastic edging, if you use that. Try to arrange the edging so that you can mow right over it and do not have to trimmer it. Every month or so, run an edger along the edging to clean it up and prevent the grass from getting into the beds.

If you hated the concrete hexagons, chances are good you won't be thrilled with an edge of round stones, etc. either - especially when grass and weeds grow between them (and they will).

Since the edge is already established, the clean trench edge can be maintained easily with the purchase of a flat edge spade. I personally like the short handled model for the many hundreds of feet of garden beds I keep tidy. The sharp delineation between bed and lawn is a finished look to me. And it'll look even better to you when your plants settle in and put out some growth.

It loooks to me like they did a crappy job. When done properly the mulch should lie flat. Notice how it is a hill instead. This will cause washing away of the mulch. It is also why it just doesn't look right. It should still be graded so it flows water away from the house. But it shouldn't just drop off like that. THose pictures tell me the landscaper didn't properly install the beds and instead just popped a bunch of plants in, covered them with mulch and dug a trench along the edge.
The correct way to do it is to dig the trench, then grade the whole bed, so it is a flat but properly graded, then install the plants and much. I see this type of bad craftmenship all the time from so called "professional landscapers". I wonder what other shortcuts they took.

Hmmmmm, I also thought the "hilly" look to the bed looked off! The cement edging in before was about 5-6" high so maybe that's why I never thought about or noticed that the bed was elevated in comparison to the lawn (I have a flower bed on the side of my house also that does not look like this at all and is at the same level as the grass border). Now that there is only a trench, it really looks odd. So do you think I can still get away with only a trench, esp. if I use a flat edge spade as suggested to clean up the edges? You're saying the bed should be graded slightly downhill but should be flat, not mounded. The more I think about it, it seems because of the slope it may only look right if I put some edging in front of that's tall enough to make the bed look even. I don't think a sloping flowerbed would look that great if that makes sense.