Heat oil in a medium skillet over medium-high heat. Add mushrooms and cook until golden, 6 to 8 mi; remove from heat. Pat artichoke hearts dry with paper towels and coarsely chop. Combine mushrooms, artichokes, sun-dried tomatoes and shredded cheese; set aside.
Heat oven to 400°F with rack in the upper third. Working with half the dough at a time, roll into an 8-by-12-inch rectangle on a lightly floured surface. Mix together the cream cheese and Italian seasoning. Spread half the cream cheese mixture on dough, leaving a 1-inch border. Sprinkle with half the mushroom mixture.
Fold dough over filling and pinch to seal the seam and ends. Place on a parchment-lined baking sheet, seam side down. Repeat with remaining dough and ingredients. Cut a few deep vents in the top of each roll and bake for 30 min; tent loosely with foil, then bake until cooked through, 10 min more.
The recipe makes two stromboli, so half it if you aren't serving a crowd. It disappears pretty fast though, so you may want the full recipe.