Obligations Remain Intact
Leaving the country doesn't end your obligation to repay your debts. Those agreements you signed with your creditors still exist. Creditors can still report your account as delinquent, even if they can't effectively collect the money after you move outside of U.S. jurisdiction. As a result, you'd have a harder time securing new domestic accounts or maintaining the credit level of existing ones.
Failure to pay your debt affects your FICO credit score. This score is used by domestic creditors to determine your risk profile. While it doesn't have as big an impact after you leave the country, a low score can make the transition overseas more difficult. You're unlikely to qualify for high-limit cards in your new home until you show lenders you have a steady source of income. If a poor credit score leaves you unable to use your domestic cards, you'll have a more difficult time managing your money.
Civil Legal Action
Moving abroad doesn't protect you from attempts to collect the debt or from legal action. Collection agencies can still attempt to contact you overseas and persuade you to pay off the amount, although your new location limits their options considerably. If your creditor files a lawsuit and serves the papers before you leave, your departure doesn't make the lawsuit go away. The suit can proceed in your absence, and if a judgment is filed against you, the creditor can seek to recover funds from any of your assets remaining in the country. However, it can't attach a judgment to your overseas holdings, because no bilateral treaties allow for that action.
While most creditors have few options after you leave the country, the U.S. government doesn't give up easily. The Department of Homeland Security maintains the Treasury Enforcement Communications System database, which identifies individuals with unpaid tax assessments who travel to the United States. The Immigration and Customs Enforcement agency can detain people whose obligations are in the TECS database when they return to the country and flag them for follow-up enforcement, such as a visit by an IRS agent.