Textbooks are important for the cultivation of the foreign culture. For example, they might include texts/passages which explain customs and traditions of the English culture or activities which ask students to compare their own traditions with the ones of the foreign culture. Moreover, the teacher can organize games which deal with matters of the English culture. Another important activity is the preparation of oral presentations by the students; these presentations might refer to English customs/traditions found by the students after personal research.
I believe that memes are a great tool for teaching cultural awareness in a fun way. The cultural aspect is embedded in a meme as they are, most of the time, part of a specific culture. Their meaning might not be grasped, if you are not part of that group. Students of English can compare memes from different countries with the ones from their own country and the task might finish with the challenge for them to create potential memes.
Some colleagues have organised an e-book that has some great ideas, too. Have a look: Book Doing Interculturality in The English Classroom
To my knowledge, there are numerous textbooks specifically designed to grow cultural aspects of L2, either implicitly or explicitly. However, if you are interested in offering a more authentic situation, I would suggest that you assign a number of tasks in which the learners have to discover the cultural differences in a collaborative manner. What is important here, in my view, is the creativity and purposefulness put into the tasks as well as the contrastive nature of the scenarios brought into the classroom.
For me, one of the most important aspects of culture instruction is including abundant opportunities for reflection on behalf of the students. Once we begin to move away from the most superficial aspects of culture (e.g., dress, food, music, etc.), we can begin to think about the 'perspectives' that these more profound elements generate within that particular culture. Coming to terms with this requires that learners engage in deep reflection. I am fond of the ACTFL's Intercultural Can-Do Statements as well as the Intercultural Reflection Tool. Both of these documents are easily accessible online with a quick Google search. They provide great guidance in terms of fostering deep reflection in your culture instruction.