I was born José Luis Martínez Cuéllar. A long name with 3 accents in Canadian terms. In Cuba it used to be a a very common name. In Spain, it still is a very common name.
I lived in Madrid, Spain, for a year and I felt great every time I saw a corner store or anything called José Luis Flowers or José Luis & Sons Shipping. By that time the usual Cuban name was more like Judith or Yassel or Yumileidi (believe it, that's the name of a Cuban athletic champion).
I prefer the traditional names.
Continuing my story, in 1993, I had to apply for a Social Insurance number and at that time, maybe still today, I could not add a space between my two surnames. What a dilemma, lose my father's Martinez or my mother's Cuéllar surname or keep both with a hyphen. Thus was born my hyphenated, non-accented surname: Martinez-Cuellar.
So many started using Cuellar as my surname and Jose as my first name, and stopped calling me Jose Luis. So Jose Cuellar is my name for Telus, Shaw, TD, HSBC, telemarketers, etc.
And the pronunciation of 'Jose Cuellar' has always been variable, from Josee Cellular to Josie Cellar and beyond...
I usually correct these events with something like 'No way Jose'. I pronounce Jose with an H-sound (Ho-seh). And the double L of Cuellar with a Y-sound (Cooeh-Yar).
As almost nobody could pronounce Jose Cuellar, I decided in my application for a license with the Real Estate Board to be called Joe Martinez for my display name (the one in advertisements and business cards). So many baseball players are called Martinez that it seems to be easily pronounced.
After trying to pronounce Joe for few days, I backtracked and requested a correction to the Board as I could not see myself being Joe and I settled with my original Canadian non-accented Jose. It has never been a problem for me to start a conversation with a 'Pronounce It Like No Way Jose' but it would be odd if I started a conversation by asking 'Joe Who? Ah! Joe the realtor'.
As you can see, I will answer to many variants of my name, though Jose Martinez might give you the most direct access to my attention.