Place 1 cup cilantro leaves, oil, 2 Tbsp. lime juice, orange juice, garlic, and 1/4 tsp. salt in a food processor or blender; process until smooth. Transfer half the mixture to a small bowl and refrigerate.
Place the remaining mixture in a sealable plastic bag and add chicken. Seal and turn to coat the chicken. Refrigerate for 30 minutes to 1 hour.
Meanwhile, combine tomatoes, strawberries, jalapeño, lime zest, and the remaining 1 Tbsp. lime juice, 1/4 cup chopped cilantro, and 1/8 tsp. salt in a medium bowl; toss well. Refrigerate until ready to serve.
Preheat grill to medium-high. Lightly oil the grill grate (see Tip). Remove the chicken from the marinade (discard the marinade). Grill the chicken until browned and cooked through, 4 to 5 minutes per side. (Alternatively, broil the chicken for 4 to 5 minutes per side.) Let the chicken rest for 5 minutes. Slice into thin strips.
To assemble: Place 1/4 cup spinach and a generous 1/3 cup chicken on each tortilla. Top each with about 1 1/2 tsp. of the reserved cilantro mixture and 1/4 cup salsa.