As mike mentioned, yes, there are methods to 'hide' the gutter hangers. Hidden Hangers can be used if the gutters are currently being supported w/spikes.
Also, consider SEAMLESS gutters. And, take a look at the 'location' of where the downspouts are located. This is a perfect time to see if putting the downspouts on a DIFFERENT end of the house is a good idea, or not. Just 'pretend' that water is pouring out the downspout - Now where will it go???
And lastly, you mentioned insulation. If the ROOF/RAFTER BAYS are insulated, and if your attic is NOT a 'finished space', that could be the cause of your icicles, or ice damming. For more on Ice Damming, here's a quick read:
<a href="http://www.grassroots.ca/homeowner_help_articles/ice_damming.htm">What is ice damming</a>.
My best to ya and hope this helps.
Jay J -Moderator
PS: Contractors for Seamless Gutters should be in the Yellow Pages under Gutters.