Hi emily,
I'm not sure I understand what you're after, but might you be looking for a contemporary finger pull? (Click on the link below and scroll down toward the bottom of the page)

Here is a link that might be useful: Finger cabinet pulls

Go to MasterMetalWorks. They make edge pulls and do some custom work, which would involve just cutting one of their options to the length you want.

Here is a link that might be useful: MMW

Carol, Pamlimpset, Thank you. I think you found them. On Palimpset's link it's called a 'bead on a radius edge.' I could also go to a metal shop here. Didn't think of having it costume made. Do you know if anyone sells them already made -- 24' long for dishwasher, and even longer for refrigerator? I can't think of anything more minimal than that.

Do you think they would work? Or, would they be too uncomfortable for daily use in an appliance?

I think you are looking for this.

Depending upon your DW and your Fridge, this may not attach in the right way depending upon how the panel is attached to the appliance doors. This type of edge pull would have interfered with my freezer drawer panel attachment.

The edge may also not have been the right place for my DW either: the door latches very firmly and this may not have given a good enough grip. You will need to talk to your cabinet maker about these.

The thicker MMW ones I showed you will also need to be mortised in to the back of the panel because of their thickness

Here is a link that might be useful: aluminum extruded edge pull

Thank you palimpset. It helps to have a picture and a name.

Emily - how did you resolve your problem ? I'm doing a black kitchen, no handles on the cabinets and a panel ready refrigerator and am also looking for a minimal appliance pullfor the refrigerator - as flat/close to the door as possible or an edge pull.