Many countries will require
you to obtain a Visa even if you become a US citizen. This is called "reciprocity ' where a country will levy a charge of x amount of dollars to obtain a Visa as its citizens are charged the same.
What is more important is if, as a PR of US, you need to obtain a Visa in advance, which means you will not be able to obtain a Visa at the airport or border crossing.
An example is Peru which requires Indian Nationals to obtain a Visa prior to travel to that country. Some countries will take 4 to 5 weeks if you are Indian, Pakistani or Bangladeshi nationals.
Hope this helps.
TPK
Switzerland and Canada don't require for PR's to have visa
Thailand will allow US PR's of certian nationalities (India included) to obtain a visa at the Airport